Since in general relativity also the size and shape of the fiber FF is dynamical, generically effective field theories arising from KK-compactification contain spurious fields parameterizing the geometry of FF. In the simplest case this is just the dilaton, encoding the total volume of FF, more generally these fields are often called the moduli fields. Since these moduli fields are not observed in experiment, naive KK-models are generically phenomenologically unviable. However, in variants of gravity such as higher dimensional supergravity there are possibilities for the moduli to obtain masses and hence for the KK-models to become viable after all. This is the problem of moduli stabilization.

Moreover, all this of course remains true if the productX×FX \times F – which we may think of as the trivial FF-fiber bundle over XX – is generalized to any associated bundleE→XE \to X with fiberFF, associated to a GG-principal bundleP→XP \to X (hence such that E=P×GFE = P \times_G F), in which case the above decomposition of the metric applies locally.

A pseudo-Riemannian manifold of this form (E,gEKK)\left(E, g^KK_{E}\right) for fixed moduligFg_F is called a Kaluza-Klein compactification of the spacetimeEE. One also speaks of the effective spacetime XX as being obtained by dimensional reduction from the spacetime EE.

may be thought of as the moduli stack of fields for an (n−k)(n-k)-dimensional field theory. By the definition universal property of the mapping stack, this lower dimensional field theory is then such that a field configiuration over an (n−k)(n-k)-dimensional spacetimeXn−kX_{n-k}

on the product space Xn−k×ΣkX_{n-k}\times \Sigma_k (the trivial Σk\Sigma_{k}-fiber bundle over Xn−kX_{n-k}).

Traditionally KK-reduction is understood as retaining only parts of Fieldsn−k\mathbf{Fields}_{n-k} (the “0-modes” of fields on Σk\Sigma_k only) but of course one may consider arbitrary corrections to this picture and eventially retain the full information.

Reductions of pure gravity with realistic gauge groups

In (Witten 81) it was observed that the minimal dimension of a fiberFF for the KK-reduction to yield the gauge groupSU(3)×SU(2)×U(1)SU(3) \times SU(2) \times U(1) is dF=7d_F = 7 . This may be a meaningless numerical coincidence, but might be – and was regarded as being – remarkable: because it means that the minimum total dimension of a KK-compactification X×FX \times F that could yield a realistic model of observed physics is 4+7=114 + 7 = 11. This is the uniquely specified dimensional of the maximal supergravity model: 11-dimensional supergravity.

Largely due to this result, the original pure Kaluza-Klein ansatz that starts with just pure Einstein gravity with no other fields) is nowadays regarded as a non-viable to produce the standard model of particle physics. But one can further play with the idea and consider more flexible models that still exhibit the essence of KK-reduction in parts.

does produce the previously missing positivepotentials for gFg_F proportional to these cycles of ℱ\mathcal{F}. So KK-reduction of 10-dimensional supergravities can – for a suitable ansatz – cure the old problem of moduli stabilization in KK-theory.

This means that physical model building using the specific ansatz of KK-reduction of type II supergravities on Calabi-Yau fibers reduces to a noteworthy enumerative problem in complex geometry: classify all real 6-dimensional Calabi-Yau manifolds with given isometries and given cycles.

While interesting, there are few tools known for performing this classification. The only thing that seems to be clear is that the classification is not sparse: there are many points in this space of choices. Since all this is relevant in model building in string theory, the space of these choices has been termed the landscape of string theory vacua.

Hence by iteratively applying KK-reductions and other dualities and topological twists, one finds long cascades of different quantum field theories that all superficially look very different, but which thereby become closely related as different aspects of one single higher dimensional field theory. Not all of these lower dimensional theories can be phenomenological viable models, but even the superficially “unrealistic” theories such as the 6d (2,0)-superconformal QFT serve, via KK-reduction, to explain and illuminate deep properties of (semi-)realistic theories such as super Yang-Mills theory in 4d.

A textbook account of the geometry behind the Lorentz force in the Kaluza-Klein mechanism (the idea that geodesics on the gauge bundle project to curved trajectories on the base manifold) can be found in the introduction of chapter 1 and in chapters 9 and 10 of: